Imagine an insulated steel rod of length 
whose ends are held at a constant 
F. Let 
be the temperature at each point along the rod at time 
, if the initial heat distribution along the rod is given by 
, find the temperature along the rod as a function of the position (
) and time ().
Let the initial heat distribution be given by the graph below.
Since no heat is being added or lost along the length of the rod, the rod will aim to achieve equilibrium at 
F. Thinking about it, the temperature will be changing most quickly at the peaks and valleys. In particular, the rate at which the temperature changes is proportional to the concavity at each point along the graph, thus the Heat Equation:
Where 
is the thermal conductivity.
The solution to the heat equation subject to the conditions above is given can be seen by clicking here.
Now how do we get there?
That will be answered in my next post.